Let $M$ be an $n$-dimensional manifold with metric $g$. The sectional curvature of the metric $g$ at a point $p\in M$ is the Gaussian curvature $K$ (at $p$) of the geodesic surface whose tangent space at $p$is spanned by vectors $X_{}$ and $Y_{}$. If $R$ is the covariant form of the curvature tensor (that is, $R$ is a tensor of type $\left(\genfrac{}{}{0ex}{}{0}{4}\right)$), then

If $K$ is independent of the choice of the vectors $X$ and $Y$ then $K\=\frac{S}{n\left(n-1\right)}$, where $S$ is the Ricci scalar of $g$.

This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form SectionalCurvature(...) only after executing the command with(DifferentialGeometry) and with(Tensor) in that order.  It can always be used in the long form DifferentialGeometry:-Tensor:- SectionalCurvature(...).

$\mathrm{with}\left(\mathrm{DifferentialGeometry}\right)\:$$\mathrm{with}\left(\mathrm{Tensor}\right)\:$

$\mathrm{DGsetup}\left(\left[x\,y\right]\,\mathrm{M1}\right)$

$\mathrm{frame\; name:\; M1}$

$\mathrm{g1}≔\mathrm{evalDG}\left(\frac{1}{y^2}\left(\mathrm{dx}\&t\mathrm{dx}+\mathrm{dy}\&t\mathrm{dy}\right)\right)$

$\mathrm{g1}:=\frac{\mathrm{dx}\mathrm{dx}}{y^2}\+\frac{\mathrm{dy}\mathrm{dy}}{y^2}$

$\mathrm{R1}≔\mathrm{CurvatureTensor}\left(\mathrm{g1}\right)$

$\mathrm{R1}:=-\frac{\mathrm{D\_x}\mathrm{dy}\mathrm{dx}\mathrm{dy}}{y^2}\+\frac{\mathrm{D\_x}\mathrm{dy}\mathrm{dy}\mathrm{dx}}{y^2}\+\frac{\mathrm{D\_y}\mathrm{dx}\mathrm{dx}\mathrm{dy}}{y^2}-\frac{\mathrm{D\_y}\mathrm{dx}\mathrm{dy}\mathrm{dx}}{y^2}$

$K≔\mathrm{SectionalCurvature}\left(\mathrm{g1}\,\mathrm{R1}\,\mathrm{D\_x}\,\mathrm{D\_y}\right)$

$K:=-1$

$R≔\mathrm{RaiseLowerIndices}\left(\mathrm{g1}\,\mathrm{R1}\,\left[1\right]\right)$

$R:=-\frac{\mathrm{dx}\mathrm{dy}\mathrm{dx}\mathrm{dy}}{y^4}\+\frac{\mathrm{dx}\mathrm{dy}\mathrm{dy}\mathrm{dx}}{y^4}\+\frac{\mathrm{dy}\mathrm{dx}\mathrm{dx}\mathrm{dy}}{y^4}-\frac{\mathrm{dy}\mathrm{dx}\mathrm{dy}\mathrm{dx}}{y^4}$

$\mathrm{R1212}≔\mathrm{Tools}:-\mathrm{DGinfo}\left(R\,''CoefficientList''\,\left[\left[1\,2\,1\,2\right]\right]\right)\left[1\right]$

$\mathrm{R1212}:=-\frac{1}{y^4}$

$\frac{\mathrm{R1212}}{\mathrm{MetricDensity}\left(\mathrm{g1}\,2\right)}$

$-1$

$\mathrm{DGsetup}\left(\left[x\,y\,z\right]\,\mathrm{M2}\right)$

$\mathrm{frame\; name:\; M2}$

$\mathrm{g2}≔\mathrm{evalDG}\left(\frac{a^2}{{\left(k^2+x^2+y^2+z^2\right)}^2}\left(\mathrm{dx}\&t\mathrm{dx}+\mathrm{dy}\&t\mathrm{dy}+\mathrm{dz}\&t\mathrm{dz}\right)\right)$

$\mathrm{g2}:=\frac{a^2\mathrm{dx}\mathrm{dx}}{{\left(k^2\+x^2\+y^2\+z^2\right)}^2}\+\frac{a^2\mathrm{dy}\mathrm{dy}}{{\left(k^2\+x^2\+y^2\+z^2\right)}^2}\+\frac{a^2\mathrm{dz}\mathrm{dz}}{{\left(k^2\+x^2\+y^2\+z^2\right)}^2}$

$X≔\mathrm{evalDG}\left(\mathrm{D\_x}+r\mathrm{D\_y}+s\mathrm{D\_y}\right)$

$X:=\mathrm{D\_x}\+\left(s\+r\right)\mathrm{D\_y}$

$Y≔\mathrm{evalDG}\left(\mathrm{D\_y}+t\mathrm{D\_z}\right)$

$Y:=\mathrm{D\_y}\+t\mathrm{D\_z}$

$\mathrm{R2}≔\mathrm{CurvatureTensor}\left(\mathrm{g2}\right)\:$

$\mathrm{K2}≔\mathrm{SectionalCurvature}\left(\mathrm{g2}\,\mathrm{R2}\,X\,Y\right)$

$\mathrm{K2}:=\frac{4k^2}{a^2}$

$\mathrm{S2}≔\mathrm{RicciScalar}\left(\mathrm{g2}\,\mathrm{R2}\right)$

$\mathrm{S2}:=\frac{24k^2}{a^2}$

$f≔\frac{a}{k^2+x^2+y^2+z^2}$

$f:=\frac{a}{k^2\+x^2\+y^2\+z^2}$

$\mathrm{FR}≔\mathrm{FrameData}\left(\left[f\mathrm{dx}\,f\mathrm{dy}\,f\mathrm{dz}\right]\,\mathrm{M3}\right)\:$

$\mathrm{DGsetup}\left(\mathrm{FR}\right)$

$\mathrm{frame\; name:\; M3}$

$\mathrm{g3}≔\mathrm{evalDG}\left(\mathrm{\Θ1}\&t\mathrm{\Θ1}+\mathrm{\Θ2}\&t\mathrm{\Θ2}+\mathrm{\Θ3}\&t\mathrm{\Θ3}\right)$

$\mathrm{g3}:=\mathrm{\Θ1}\mathrm{\Θ1}\+\mathrm{\Θ2}\mathrm{\Θ2}\+\mathrm{\Θ3}\mathrm{\Θ3}$

$\mathrm{R3}≔\mathrm{CurvatureTensor}\left(\mathrm{g3}\right)\:$

$\mathrm{K3}≔\mathrm{SectionalCurvature}\left(\mathrm{g3}\,\mathrm{R3}\,\mathrm{E1}+r\mathrm{E2}+t\mathrm{E3}\,\mathrm{E2}+t\mathrm{E3}\right)$

$\mathrm{K3}:=\frac{4k^2}{a^2}$

$\mathrm{DGsetup}\left(\left[x\,y\,z\right]\,\mathrm{M4}\right)$

$\mathrm{frame\; name:\; M4}$

$\mathrm{g4}≔\mathrm{evalDG}\left(y\mathrm{dx}\&t\mathrm{dx}+\mathrm{dy}\&t\mathrm{dy}+\mathrm{dz}\&t\mathrm{dz}\right)$

$\mathrm{g4}:=y\mathrm{dx}\mathrm{dx}\+\mathrm{dy}\mathrm{dy}\+\mathrm{dz}\mathrm{dz}$

$X≔\mathrm{evalDG}\left(\mathrm{D\_x}+r\mathrm{D\_y}+s\mathrm{D\_z}\right)$

$X:=\mathrm{D\_x}\+r\mathrm{D\_y}\+s\mathrm{D\_z}$

$Y≔\mathrm{evalDG}\left(\mathrm{D\_y}+t\mathrm{D\_z}\right)$

$Y:=\mathrm{D\_y}\+t\mathrm{D\_z}$

$\mathrm{R4}≔\mathrm{CurvatureTensor}\left(\mathrm{g4}\right)\:$

$\mathrm{K4}≔\mathrm{SectionalCurvature}\left(\mathrm{g4}\,\mathrm{R4}\,X\,Y\right)$

$\mathrm{K4}:=\frac{1}{4y\left(s^2\+t^2{r}^2\+y{t}^2\+y-2tsr\right)}$